Toeplitz Quotient C*-Algebras and Ratio Limits for Random Walks

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We study quotients of the Toeplitz C*-algebra of a random walk, similar to those studied by the author and Markiewicz for finite stochastic matrices. We introduce a new Cuntz-type quotient C*-algebra for random walks that have convergent ratios of transition probabilities. These C*-algebras give rise to new notions of ratio limit space and boundary for such random walks, which are computed by appealing to a companion paper by Woess.

Original languageEnglish
Pages (from-to)1529-1556
Number of pages28
JournalDocumenta Mathematica
StatePublished - 2021
Externally publishedYes

Bibliographical note

Funding Information:
The author is grateful to Wolfgang Woess for many helpful exchanges on the subject of random walks and their boundaries, for providing remarks on this paper, and for computing ratio limit boundaries in many classes of examples in the companion paper [58]. The author is also grateful to Christopher Linden and Alex Vernik for suggestions, discussions and remarks on draft versions of this paper. The author was partially supported by NSF grant DMS-1900916 and by the European Union’s Horizon 2020 Marie Sklodowska-Curie grant No 839412.

Publisher Copyright:
© 2021, Documenta Mathematica.All Rights Reserved.


  • Cuntz algebras
  • Gauge-invariant uniqueness
  • Martin boundary
  • Random walks
  • Ratio limit boundary
  • Strong ratio limit property
  • Subproduct systems
  • Symmetry equivariance
  • Toeplitz quotients

ASJC Scopus subject areas

  • Mathematics (all)


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